That is actually a Satellaview game.
If you don't know, Satellaview was a satellite add-on released in Japan that allowed users to download games. Games like this and the two exclusive Zelda games were meant to be played with audio streamed from the satellite-radio network (thus the actual gameplay was a bit limited as it had to regularly delay the player's ability to progress to sync with the radio broadcast).
Though there were a number of games released for Satellaview that could be played offline, which is why the hardware is now somewhat sought after.
From watching recorded videos, some other games just played random pop music during the game, so even if Nintendo had the original recordings they'd still have licensing issues, unless they want to replace the music or just leave the game oddly silent. (as amusing as it is to listen to A View To A Kill during All-Stars Super Mario Bros. I thought I remember the Power of Love being in the video too, but Mario seems to be lacking in that department ).